Recovering the initial state of a Well-Posed Linear System with skew-adjoint generator

We consider the problem of recovering the initial data of a Well-Posed Linear System with unitary semigroups. It is well-known that this inverse problem is well posed if the system is exactly observable, but this assumption may be very restrictive in some applications. Thus, we are interested in systems which are not exactly observable, and in particular, where we cannot expect a full reconstruction. We propose to use the algorithm studied by Ramdani et al. in (Automatica 46:1616–1625, 2010) and prove that it always converges towards the observable part of the initial state

Analytic Controllability of Time-Dependent Quantum Control Systems

The question of controllability is investigated for a quantum control system in which the Hamiltonian operator components carry explicit time dependence which is not under the control of an external agent. We consider the general situation in which the state moves in an infinite-dimensional Hilbert space, a drift term is present, and the operators driving the state evolution may be unbounded. However, considerations are restricted by the assumption that there exists an analytic domain, dense in the state space, on which solutions of the controlled Schrodinger equation may be expressed globally in exponential form. The issue of controllability then naturally focuses on the ability to steer the quantum state on a finite-dimensional submanifold of the unit sphere in Hilbert space -- and thus on analytic controllability. A relatively straightforward strategy allows the extension of Lie-algebraic conditions for strong analytic controllability derived earlier for the simpler, time-independent system in which the drift Hamiltonian and the interaction Hamiltonia have no intrinsic time dependence. Enlarging the state space by one dimension corresponding to the time variable, we construct an augmented control system that can be treated as time-independent. Methods developed by Kunita can then be implemented to establish controllability conditions for the one-dimension-reduced system defined by the original time-dependent Schrodinger control problem. The applicability of the resulting theorem is illustrated with selected examples.Comment: 13 page

Recovering the observable part of the initial data of an infinite-dimensional linear system with skew-adjoint generator

We consider the problem of recovering the initial data (or initial state) of infinite-dimensional linear systems with unitary semigroups. It is well-known that this inverse problem is well posed if the system is exactly observable, but this assumption may be very restrictive in some applications. In this paper we are interested in systems which are not exactly observable, and in particular, where we cannot expect a full reconstruction. We propose to use the algorithm studied by Ramdani et al. in (Automatica 46:1616–1625, 2010) and prove that it always converges towards the observable part of the initial state. We give necessary and sufficient condition to have an exponential rate of convergence. Numerical simulations are presented to illustratethe theoretical results

Stabilization of linear time-varying systems

For linear time-varying systems with bounded system matrices we discuss the problem of stabilizability by linear state feedback. It is shown that an optimal control approach yields a criterion in terms of the cost for stabilizability. The constants appearing in the criterion of optimality allow for the distinction of exponential and uniform exponential stabilizability. We show that the system is completely controllable if, and only if, the Lyapunov exponent is arbitrarily assignable by a suitable feedback. For uniform exponential stabilizability and the assignability of the Bohl exponent this property is known. Also, dynamic feedback does not provide more freedom to address the stabilization problem

Infinite-dimensional Lur'e systems with almost periodic forcing

We consider forced Lur’e systems in which the linear dynamic component is an infinite-dimensional well-posed system. Numerous physically motivated delay- and partial-differential equa-tions are known to belong to this class of infinite-dimensional systems. We present refinements ofrecent incremental input-to-state stability results [14] and use them to derive convergence results fortrajectories generated by Stepanov almost periodic inputs. In particular, we show that the incrementalstability conditions guarantee that for every Stepanov almost periodic input there exists a unique pairof state and output signals which are almost periodic and Stepanov almost periodic, respectively. Thealmost periods of the state and output signals are shown to be closely related to the almost periodsof the input, and a natural module containment result is established. All state and output signalsgenerated by the same Stepanov almost periodic input approach the almost periodic state and theStepanov almost periodic output in a suitable sense, respectively, as time goes to infinity. The sufficientconditions guaranteeing incremental input-to-state stability and the existence of almost periodic stateand Stepanov almost periodic output signals are reminiscent of the conditions featuring in well-knownabsolute stability criteria such as the complex Aizerman conjecture and the circle criterion

How to get a conservative well-posed linear system out of thin air. Part I. Well-posedness and energy balance

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Stabilization of Time-Varying System by Controllers with Internal Loop

We study the concept of stabilization with internal loop for infinite-dimensional discrete time-varying systems in the framework of nest algebra. We originally give a parametrization of all stabilizing controllers with internal loop, and it covers the parametrization of canonical or dual canonical controllers with internal loop obtained before. We show that, in practical application, the controller with internal loop overcomes the awkwardness brought by the extra invertibility condition in the parametrization of the conventional controllers. We also prove that the strong stabilization problem can be completely solved in the closed-loop system with internal loop. Thus the advantage of the controller with internal loop is addressed in the framework of nest algebra